The lectures are usually given every second week at 3:00 pm Thursdays in 103 McAllister. Coffee at 2:30 pm in 103 McAllister.

Thur, Sep 2 Greg Swiatek (Penn State)
3:00 pm The Meaning of Chaos
Thur, Sep 23 Dale Brownawell (Penn State)
3:00 pm When Do Polynomials Have Common Zeros?
Abstract: The answer is easy when the polynomials involve only one
variable or when they are linear. The elegant general
answer is still being investigated more than 100 years
after its discovery.
Related questions are:
How hard can this question be, anyway?
How small can polynomial values be?
These are active areas of investigation.
Thur, Oct 14 Arkady Vaintrob (University of New Mexico)
3:00 pm Knots and Knot Invariants
Abstract: A systematic study of knots was initiated by physicists
in the second half of the last century. In the beginning
of this century mathematicians took over the field and
introduced knot invariants - tools for comparing different
knot types. Near the end of the century physicists struck
back by inventing a host of new more powerful invariants
whose geometric meaning, however, remained mysterious.
Recently as a result of joint efforts of physicists and
mathematicians, unexpected connections between old and
new knot invariants have been discovered raising
expectations for new exciting results to come.
In the lecture, after a brief introduction to knot theory,
we will discuss the current developments by comparing the
classical Alexander invariant and the most famous
invariant of the new era, the Jones polynomial.
Thur, Oct 21 Yulij Ilyashenko (Cornell)
3:00 pm Hilbert's 16th Problem Near Its Centenary
Abstract: The second part of the 16th problem appeared to be one
of the most difficult in the Hilbert's list. Smale
included it in his list of the main problems for the
forthcoming century. In the expiring century the problem
had a long and dramatic history. In the last 20 years
interest in the problem has grown substantially, and
some progress was achieved. Several filial problems
about "Hilbert type numbers" were stated and partially
solved. The subject of the talk will be a sketch of this
history and achievements.
Tues, Oct 26 Viorel Nitica (Notre Dame)
3:00 pm Replicating Tiles
Abstract: A plane figure (or tile) is defined to be replicating
of order k (or rep-k) if it can be dissected into k
replicas, each congruent to the other and similar to
the original. If k=4, an equivalent formulation is that
four identical figures are to be assembled into a scale
model, twice as long and twice as high. All triangles
and parallelograms are rep-4 tiles.
I shall start my lecture by showing a list of rep-k
tiles for various values of k. Then I shall relate the
study of rep-tiles to more general questions that are
subject to current research, such as when can a finite
region consisting of cells in a square lattice be
perfectly tiled by tiles drawn from a finite set of
shapes?
Thur, Nov 4 Yuri Latushkin (University of Missouri)
3:00 pm Nightmares and Dreams of Lyapunov: Stability and Semigroups
Abstract: After a five-minute mini-course in functional analysis,
we will discuss how semigroups of linear operators are
related to the stability of linear differential equations
in infinite dimensional spaces. A classical theorem of A.
M. Lyapunov says that the equation is stable provided the
spectrum of its (bounded) coefficient belongs to the open
left half-plane. If the coefficient is an unbounded
operator, then this theorem, generally, does not hold. We
will discuss a replacement of the Lyapunov Theorem that
uses so-called evolution semigroups.
Thur, Nov 11 Alexander Dranishnikov (Penn State)
3:00 pm On the Hilbert-Smith Conjecture
Abstract: The Hilbert-Smith conjecture is a remaining branch of the
fifth Hilbert problem. We will discuss how it leads to
p-adic numbers and strange phenomena in dimension theory.
Thur, Nov 18 Serge Tabachnikov (University of Arkansas)
3:00 pm The DNA Geometric Inequality: An Open-Ended Story
Abstract: Consider a closed plane curve, possibly self-intersecting,
inside a closed convex plane curve; the former curve is
referred to as DNA and the latter as Cell.
The DNA geometric inequality asserts that the average
absolute curvature of DNA is not less than that of Cell.
I will discuss this simple result (surprisingly, proved
only a few years ago) and give five different proofs of
the particular case when Cell is a circle.
Thur, Dec 2 Misha Guysinsky (Tufts University)
3:00 pm The Banach-Tarski Paradox and Amenable Groups
Abstract: Is it possible to cut up a pea into finitely many pieces
that can be rearranged to form a ball the size of the
sun? The answer to this question looks obvious, but it is
not. We define a special class of groups called amenable
groups and discuss its connection with questions like
this.
Conrad Plaut (University of Tennessee)
3:00 pm Group Reconstruction
Abstract: What happens when you reconstruct a group using only
information from a small piece of the group? In 1928
O. Schreier discovered a simple way to do such reconstuction,
which was rediscovered later by A. Mal'tsev, and later
still by J. Tits. We begin with a little known and
disastrous solution to Hilbert's Fifth Problem that was
published in the Annals of Mathematics in 1957. We then
go on to show how Schreier's fundamental construction is
connected to covering group theory and finitely represented
froups.